Clique immersion in graph products

TL;DR

This paper investigates how the maximum clique immersion number behaves under various graph products, providing bounds and conjectures for different types of products.

Contribution

It establishes best possible lower bounds for clique immersion numbers in Cartesian and lexicographic products and proposes conjectures for direct and strong products.

Findings

Lower bounds for Cartesian and lexicographic products

Conjectures for direct and strong products

Partial results supporting the conjectures

Abstract

Let $G,H$ be graphs and $G*H$ represent a particular graph product of $G$ and $H$. We define $im(G)$ to be the largest $t$ such that $G$ has a $K_t$-immersion and ask: given $im(G)=t$ and $im(H)=r$, how large is $im(G*H)$? Best possible lower bounds are provided when $*$ is the Cartesian or lexicographic product, and a conjecture is offered for each of the direct and strong products, along with some partial results.